# How do you find the area bounded by the cardioid r=1+cos(theta)?

Mar 27, 2016

$A = \frac{\left(\cos x + 4\right) \sin x + 3 x}{4}$

#### Explanation:

Given: $r = 1 + \cos \left(\theta\right)$
Required: Area of cardioid?
Solution Strategy: Polar Coordinate Area Integral
$A = {\int}_{{\theta}_{1}}^{{\theta}_{2}} \frac{1}{2} {r}^{2} d \left(\theta\right)$ substitute for $r$
$A = \frac{1}{2} {\int}_{{\theta}_{1}}^{{\theta}_{2}} {\left(1 + \cos \left(\theta\right)\right)}^{2} d \left(\theta\right)$
$= \frac{1}{2} \left[\int \left(1 + 2 \cos \theta + {\cos}^{2} \theta\right) d \left(\theta\right)\right]$
$= \frac{1}{2} \left[\theta + 2 \sin \theta + \int {\cos}^{2} \theta d \left(\theta\right)\right]$
${I}_{3} = \textcolor{b r o w n}{\int {\cos}^{2} \theta d \left(\theta\right)}$ apply reduction formula
${I}_{3} = \frac{n - 1}{n} \int {\cos}^{n - 2} \theta d \left(\theta\right) + \frac{{\cos}^{n - 1} \theta \sin \theta}{n}$
${I}_{3} = \textcolor{b r o w n}{\frac{1}{2} \theta + \frac{\cos \theta \sin \theta}{2}}$
Putting it all together:

$A = \frac{\left(\cos x + 4\right) \sin x + 3 x}{4}$